• In all previous examples, we can achieve good to excellent predictions
• But we do not merely want to predict systems, but also change them!

• Teaser:

\[ \begin{aligned} X &:= \epsilon_X \\[.5em] Y &:= X + \epsilon_Y \\[.5em] Z &:= Y + \epsilon_Z \enspace , \end{aligned} \]

• with \(\epsilon_X, \epsilon_Y \stackrel{\text{iid}}{\sim} \mathcal{N}(0, 1)\) and \(\epsilon_Z \stackrel{\text{iid}}{\sim} \mathcal{N}(0, 0.1)\)

• Does \(X\) or \(Z\) predict \(Y\) better?

• 350 patients chose to take a drug and 350 chose not to
• Given these data, should a doctor prescribe the drug to a patient?

• 1) Motivation
  • Spurious Correlations
  • Modeling for Interventions

• 2) Causal Inference
  • Seeing versus Doing
  • Structural Causal Models
  • Causal Effects and Confounding

• 3) Simpson’s Paradox

• 4) Exercises

• An introduction to Causal inference is based on this talk

• Directed Acyclic Graphs (DAGs) visualize (conditional) independencies

• d-separation allows us to read off (conditional) independencies from any DAG
• We need to define a few concepts first:
  • A path \(\omega\) from \(X\) to \(Y\) is a sequence of nodes & edges such that the start & end nodes are \(X\) and \(Y\)
  • A conditioning set \(\mathcal{L}\) is the set of nodes we condition on (it can be empty)
  • Conditioning on a collider or any of its descendants unblocks a path (e.g., \(X \rightarrow Z \leftarrow Y\))

• Two nodes \(X\) and \(Y\) are d-separated by \(\mathcal{L}\) if and only if members of \(\mathcal{L}\) block all paths between \(X\) and \(Y\)

• Number of storks \(X\) are independent of birth rate \(Y\) given environmental development \(Z\)
  • This can be visualized in three different, Markov equivalent DAGs!
  • Doing could mean for example setting \(Z\) to some value \(z\) (hard intervention)

• Structural Causal Models (SCM) as the fundamental building blocks of causal inference
  • We understand relations between variables in a SCM to be causal
  • Here, we will assume acyclic, linear SCMs with independent Gaussian error terms
  • The relations in an SCM can be visualized in a directed acyclic graph

• Example:

\[ \begin{aligned} X &:= \epsilon_X \\[.5em] Y &:= X + \epsilon_Y \\[.5em] Z &:= Y + \epsilon_Z \enspace , \end{aligned} \]

• with \(\epsilon_X, \epsilon_Y \stackrel{\text{iid}}{\sim} \mathcal{N}(0, 1)\) and \(\epsilon_Z \stackrel{\text{iid}}{\sim} \mathcal{N}(0, 0.1)\)

• Note the factorization of the joint distribution:

\[ p(X_1, \ldots, X_n) = \prod_{i=1}^n p(X_i \mid \text{Parents}(X_i)) \hspace{6em} p(X, Y, Z) = p(Z \mid Y) \, p(Y \mid X) \, p(X) \]

set.seed(1)

n <- 100
x <- rnorm(n, 0, 1)
y <- x + rnorm(n, 0, 1)
z <- y + rnorm(n, 0, 0.1)

• Interventional distribution \(p(Y \mid do(Z))\) is observational distribution \(p_m(Y \mid Z)\) in manipulated DAG
• For linear, acyclic SCMs we define the average causal effect:

\[ ACE(Z \rightarrow Y) = \mathbb{E}\left[Y \mid do(Z = z + 1) \right] - \mathbb{E}\left[Y \mid do(Z = z) \right] \enspace . \]

• Confounding: The causal effect of \(Z\) on \(Y\) is confounded if \(p(Y \mid Z) \neq p(Y \mid do(Z))\).

set.seed(1)

get_ce <- function(zvalue) {
  n <- 100
  x <- rnorm(n, 0, 1)
  y <- x + rnorm(n, 0, 1)
  z <- zvalue
  
  mean(y) # E[Y | do(Z = zvalue)]
}

(ACE <- get_ce(1) - get_ce(0))

## [1] -0.0101961

• From the SCM, we see that the \(ACE(X \rightarrow Y) = 1\)
• This corresponds to what we find by intervening on \(X\):

set.seed(1)

get_ce <- function(xvalue) {
  n <- 100
  x <- xvalue
  y <- x + rnorm(n, 0, 1)
  z <- rnorm(n, 0, 0.1)
  
  mean(y) # E[Y | do(X = xvalue)]
}

(ACE <- get_ce(1) - get_ce(0))

## [1] 1.079214

• In observational data, there will always be confounding factors
  • What variables should we adjust for?
  • This requires knowledge about the underlying DAG
  • (Don’t just adjust for all variables — that can induce bias!)

• Backdoor Criterion (Pearl, Glymour, & Jewell, 2016, p. 61): An adjustment set \(\mathcal{Z}\) fulfills the backdoor criterion if no member in \(\mathcal{Z}\) is a descendant of \(X\) and members in \(\mathcal{Z}\) block all paths between \(X\) and \(Y\). Adjusting for \(\mathcal{Z}\) thus yields the causal effect of \(X \rightarrow Y\).

• Rationale:
  • We block all spurious paths between \(X\) and \(Y\)
  • We leave all directed paths from \(X\) to \(Y\) unperturbed
  • We create no new spurious paths

• There is a causal hierarchy:
  • Seeing: Directed Acyclic Graphs (DAGs) encode conditional independencies
  • Doing: causal DAGs encode interventional distributions (do-operator)

• In observational settings, confounding variables abound
• Causal DAGs allow us to derive valid adjustment sets

• Causal inference goes beyond prediction by modeling the outcome of interventions
• Structural Causal Models (SCMs) are the building block of (this flavour) causal inference
  • They encode observational as well as interventional distributions

• Free Causal Diagrams course by Miguel Hernán on edX

• 350 patients chose to take a drug and 350 chose not to
• Given these data, is the drug helpful or harmful?
• Should a doctor prescribe the drug to a patient?

• 350 patients chose to take a drug and 350 chose not to
• Given these data, is the drug helpful or harmful?
• Should a doctor prescribe the drug to a patient?

• The data are exactly the same in both cases
• Statistics alone cannot provide an answer
• We need to have an understanding of the causal mechanism behind these data
• We can visualize this causal mechanism using directed acyclic graphs (DAGs)

• Suppose we know that estrogen has a negative effect on recovery
• Note also that more women choose the drug than men
• Therefore, being a woman has an effect on drug taking as well as recovery
  • Should condition on gender!
  • This blocks the backdoor path \(D \leftarrow G \rightarrow R\)
  • And therefore unconfounds the effect \(D \rightarrow R\)

• Blood pressure is measured after taking the drug
  • It cannot cause choosing the drug
  • Instead, it is a mechanism of how the drug works
  • Should not condition on blood pressure!

• The previous examples used observational data
• Randomized control trials (RCTs) allow causal statements because they remove confounding
• However, many real life examples do not allow for RCTs
  • In theory, we can make causal statements even from observational data
  • Causal graphs can help us navigate this mess
  • In practice, difficult to arrive at the “correct” causal graph

• There is a causal hierarchy:
  • Seeing: Directed Acyclic Graphs (DAGs) encode conditional independencies
  • Doing: causal DAGs encode interventional distributions (do-operator)

• In observational settings, confounding variables abound
• Causal DAGs allow us to derive valid adjustment sets

• Causal inference goes beyond prediction by modeling the outcome of interventions
• Structural Causal Models (SCMs) are the building block of (this flavour) causal inference
  • They encode observational as well as interventional distributions

• Free Causal Diagrams course by Miguel Hernán on edX

• I blog about statistics once a month at https://fabiandablander.com/. Topics include:
  • An introduction to Causal inference
  • Bayesian modeling using Stan: A case study
  • Love affairs and linear differential equations
  • Harry Potter and the Power of Bayesian Constrained Inference
  • …

• CorrelAidxNL connects people with data science skills and NGOs
  • facebook.com/CorrelAidXNetherlands/

• Come up with two different examples for each of these graphs

• List all marginal as well as conditional independencies shown in this graph

• Draw the DAG induced by this Structural Causal Model:

set.seed(1)

n <- 1000
x <- rnorm(n, 0, 1)
y <- rnorm(n, 0, 1)
z <- x + y + rnorm(n, 0, 1)

• Can you explain what happens here?

coef(lm(y ~ 0 + x))

##           x 
## 0.006608738

coef(lm(y ~ 0 + x + z))

##          x          z 
## -0.5262511  0.5047807

• Draw the DAG induced by this SCM:

\[ \begin{aligned} S &:= \epsilon_S \\[.5em] T &:= S + \epsilon_T \\[.5em] U &:= W + \epsilon_U \\[.5em] V &:= 0.90 Z + \epsilon_V \\[.5em] W &:= 0.50 S + 1 X + 3 Y + \epsilon_W \\[.5em] X &:= Z + \epsilon_X \\[.5em] Y &:= 1.50V + X + \epsilon_Y \\[.5em] Z &:= \epsilon_Z \end{aligned} \]

• where

\[ (\epsilon_S, \epsilon_T, \epsilon_U, \epsilon_V, \epsilon_W, \epsilon_X, \epsilon_Y, \epsilon_Z) \stackrel{iid}{\sim} \mathcal{N}(0, 1) \]

• Using the previously drawn DAG, mark the following statement as either correct or incorrect:
  • Regressing \(T\) onto \(Y\) yields no effect (Note: \(T\) onto \(Y\) means ‘lm(t ~ y)’)
  • Regressing \(T\) onto \(Y\) while adjusting for \(W\) yields no effect
  • \(W\) and \(T\) are marginally independent
  • \(Y\) is a collider on the path from \(X\) to \(U\) that goes through \(Y\)
  • \(Y\) is independent of \(Z\) given given \(X\) and \(V\)
  • \(S\) is independent of \(Y\) given \(U\)
  • \(T\) and \(U\) are marginally independent; \(T\) and \(Y\) are marginally independent
  • \(X\) and \(S\) are marginally independent
  • \(X\) is independent of \(U\) given \(W\); \(X\) is independent of \(V\) given \(Z\)
  • \(X\) is independent of \(V\) given \(Z\) and \(U\)
• Generate \(n = 2000\) observations from the previous DAG
  • Check your above answers by running the relevant regressions / conditional independence tests
  • You can use the ‘ci.test’ function from the ‘bnlearn’ package

• Use the previous DAG for these questions
• Compute \(\mathbb{E}\left[W \mid X = x\right]\)
  • Does it correspond to the causal quantity in the SCM? Why / why not?

• Compute \(\mathbb{E}\left[W \mid do(X = x)\right]\)

• What is the valid adjustment set to compute the causal effect \(\mathbb{E}\left[U \mid do(Z)\right]\)?

• Visualize the marginal distribution of \(W\) under the interventional DAG \(Y:= 2\)
  • Compare it with the marginal distribution under the observational DAG
  • What do you observe? Why?

• Implement the intervention \(Y:= y\) for \(y \in [0, 4]\)
  • What do you expect \(\mathbb{E}\left[W \mid do(Y = y)\right]\) will look like as a function of \(y\)? Why?
  • What do you expect \(\text{Var}\left[W \mid do(Y = y)\right]\) will look like as a function of \(y\)? Why?
  • Visualize both!

• Download the (made-up) observational data set about mental health, exercise, and age from here.

• Visualize and analyze the data. What do you observe?

• Draw a DAG that could underlie these data. Which analysis is the correct one?